Substitute Wood headed in Johann Berg Gudmundsson's corner for the victor while Everton skipper Ashley Williams was sent off four minutes from time for a swipe at Ashley Barnes, who cancelled out Cenk Tosun's first-half opener for the visitors.
The Toffees were leading 1-0 after the first half, but while their opponents completed a first ever winning comeback in the Premier League under Sean Dych there were boos from the away end after the final whistle.
Although Pickford made a string of fine stops in front of watching England chief Gareth Southgate, he couldn't keep out Wood's victor as Everton wilted.
A disappointing afternoon for Everton worsened late on when centre-back Ashley Williams was sent off for elbowing Barnes, the Wales global now facing a suspension with Sam Allardyce's men still having work to do to make absolutely sure of safety. Click any of these buttons below!
Sensing, perhaps, that his side could hurt the visitors' shaky defence with more height up front, Dyche introduced Wood at half time and he wasted no time in hassling ex-Claret Michael Keane into conceding a corner that caused momentary panic in Everton's penalty area.
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Bortles, 25, put up the best performance of an otherwise up-and-down career in 2017. I also thought Bortles would be better off resuming his career on another team.
The win consolidated Bunrnley's seventh-place status, putting them five points behind Arsenal.
The Everton manager will have been provided with a boost in terms of players available for selection ahead of the trip, with Leighton Baines being ready to make an appearance.
Allardyce has guided Everton from the brink of the relegation zone to the top ten in the Premier League table but there has been speculation he could leave the club at the end of the season. Deservingly, the Merseyside club managed to take a 1-0 lead going into the second half but were blown away by a spirited Burnley in the later 45 minutes.
"Burnley hadn't won in 11 and if we went 2-0 up it would have been hugely hard for them".
Everton's on-loan defender Eliaquim Mangala is likely to miss the rest of the season with a knee injury, manager Sam Allardyce said on Friday.
The Clarets responded superbly, forcing Jordan Pickford into three excellent saves before Ashley Barnes equalised by converting Matthew Lowton's immaculate through ball from deep.